The Willmore energy of a closed surface in ${\mathbb{R}}^{n}$ is the integral of its squared mean curvature, and is invariant under Möbius transformations of ${\mathbb{R}}^{n}$. We show that any torus in ${\mathbb{R}}^{3}$ with energy at most $8\pi -\delta $ has a representative under the Möbius action for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on $\delta >0$. An analogous estimate is also obtained for closed, orientable surfaces of fixed genus $p\ge 1$ in ${\mathbb{R}}^{3}$ or ${\mathbb{R}}^{4}$, assuming suitable energy bounds which are sharp for $n=3$. Moreover, the conformal type is controlled in terms of the energy bounds.

Classification: 53A05, 53A30, 53C21, 49Q15

@article{ASNSP_2012_5_11_3_605_0, author = {Kuwert, Ernst and Sch\"atzle, Reiner}, title = {Closed surfaces with bounds on their Willmore energy}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 11}, number = {3}, year = {2012}, pages = {605-634}, zbl = {1260.53027}, mrnumber = {3059839}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2012_5_11_3_605_0} }

Kuwert, Ernst; Schätzle, Reiner. Closed surfaces with bounds on their Willmore energy. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 11 (2012) no. 3, pp. 605-634. http://www.numdam.org/item/ASNSP_2012_5_11_3_605_0/

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